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\title{About Kusner's Dihedral Symmetry Surface}
\author{}
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\vskip -60pt
Pseudo-Code for ``Kusner's Dihedral Symmetry Surfaces'':

Let $p =  \max( \mathrm{Round}(ee), 2)$  \hfil\break
 and let   \hfil\break

$\Re(z) = u \cos(v)$  \hfil\break
$\Im(z) = u \sin(v)$ \hfil\break
Then $P(z) = \Re( a(z) V(z) ) + (0, 0, aa)$  \hfil\break
where $a(z)$ is the complex number  \hfil\break

\[
a(z) = \frac{1}{
(z^p - z^{-p} + (2/(p-1)) \sqrt{(2p - 1)} )
}
\]

and $V(z)$ is the complex vector
\[
V(z) = ( 
i ( z^{p-1} - z^{1-p} ) ,
 z^{p-1} + z^{1-p} , 
 (i(p-1)/p) ( z^p + z^{-p} ) 
 )
\]  
 
  This is a minimal surface with dihedral symmetry of order $2p$ if $p$ is 
odd and $4p$ if $p$ is even.  
  The default value of ee is 4.  This gives the inversion in the unit sphere of 
the ``Morin Sphere Eversion Midpoint'' Willmore surface (see the surface 
surface menu).  On the other hand, when ee = 3 this gives the Inverted Boy's 
Surface (on this menu).
\hfil\break
   For full details, see:

R.  Kusner, Conformal Geometry and Complete Minimal Surfaces,
   Bulletin of the AMS, v.17, Number 2, October 1987, pp291--295.

\vfil

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